how do you factor x^2+29xy+100y^2
Use the quadratic equation, with a = 1, b = 29 y and c = 100y^2. This will give you two roots that will have "y" in them. Call them y1 and y2.
The factors will be (x-y1)(x-y2)
you know your factors must look like
(x + ?y)(x + ?y)
what two numbers add to 29 and multiply to get 100 ?
how about 25 and 4
so
(x+25y)(x+4y)
To factor the expression x^2 + 29xy + 100y^2, you need to find two binomials whose product results in the given expression. Let's break down the process step by step:
Step 1: Identify the form
The given expression is in the form of a quadratic polynomial. It cannot be factored further by using simple numeric methods like factoring out a common term or using the difference of squares formula.
Step 2: Look for patterns
The given expression has a squared term (x^2) and a constant term (100y^2). Additionally, the coefficient of the xy term (29xy) is not a perfect square. So, we can conclude that the expression cannot be factored using standard methods.
Step 3: Use the quadratic formula
Since the expression does not fit any factoring patterns, we can use the quadratic formula to find the roots of the quadratic equation. The quadratic formula states that for an equation in the form of ax^2 + bx + c = 0, the roots (or factors) can be found using the following formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = 29y, and c = 100y^2. Plugging these values into the quadratic formula, we get:
x = (-29y ± √((29y)^2 - 4(1)(100y^2))) / (2(1))
Simplifying further:
x = (-29y ± √(841y^2 - 400y^2)) / 2
x = (-29y ± √(441y^2)) / 2
x = (-29y ± 21y) / 2
This gives us two possibilities for factoring the expression:
x = (-29y + 21y) / 2 = -4y / 2 = -2y
x = (-29y - 21y) / 2 = -50y / 2 = -25y
Therefore, the factored form of x^2 + 29xy + 100y^2 is (x - 2y)(x - 25y).